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Unit 3 – RDF(S) & OWL Semantics Unit 3 – Beyond Simple Entailment: Semantics of the RDF(S), Datatypes, and OWL vocabulary Axel Polleres Siemens AG Österreich VU 184.729 Semantic Web Technologies A. Polleres VU 184.729 1/43

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Page 1: Unit3–BeyondSimpleEntailment: Semanticsofthe RDF(S ...polleres/teaching/... · Unit3–RDF(S)&OWLSemantics Unit3–BeyondSimpleEntailment: Semanticsofthe RDF(S),Datatypes,andOWLvocabulary

Unit 3 – RDF(S) & OWL Semantics

Unit 3 – Beyond Simple Entailment: Semantics of theRDF(S), Datatypes, and OWL vocabulary

Axel Polleres

Siemens AG Österreich

VU 184.729 Semantic Web Technologies

A. Polleres VU 184.729 1/43

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Unit 3 – RDF(S) & OWL Semantics

Unit Outline

1. RDF(S) Entailment: Giving semantics to the rdf: and rdfs: Vocabulary

2. D-Entailment: Giving Semantics to Datatypes

3. OWL Entailment: Giving Semantics to the owl: Vocabulary

A. Polleres VU 184.729 2/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment

Unit Outline

1. RDF(S) Entailment: Giving semantics to the rdf: and rdfs: Vocabulary

2. D-Entailment: Giving Semantics to Datatypes

3. OWL Entailment: Giving Semantics to the owl: Vocabulary

A. Polleres VU 184.729 3/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment

In the following, we use the following namespace prefixes:

@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .@prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#> .@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .@prefix owl: <http://www.w3.org/2002/07/owl#> .

A. Polleres VU 184.729 4/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

The rdf: vocabulary

The RDF vocabulary, rdfV, is a set of URI references in the rdf:namespace:

rdf:type . . . expresses class membershiprdf:Property . . . the class of propertiesrdf:XMLLiteral . . . the datatype of XML Literals.rdf:first, rdf:rest, rdf:nil, rdf:List . . . vocabulary to write lists(called “collections”) in RDFrdf:Seq, rdf:Bag, rdf:Alt, rdf:_1, rdf:_2 . . . vocabulary to describe(ordered, unordered, alternate) containers of values.rdf:Statement, rdf:subject, rdf:predicate, rdf:object. . . vocabulary to make statements about statement (aka “reification”)rdf:value . . . to describe structured values

Note: The first three are the most important for the formal semntics. Thesemantics doesn’t make any strong restrictions on the grayed out vocabulary.

A. Polleres VU 184.729 5/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

The rdf: vocabulary

The RDF vocabulary, rdfV, is a set of URI references in the rdf:namespace:

rdf:type . . . expresses class membershiprdf:Property . . . the class of propertiesrdf:XMLLiteral . . . the datatype of XML Literals.rdf:first, rdf:rest, rdf:nil, rdf:List . . . vocabulary to write lists(called “collections”) in RDFrdf:Seq, rdf:Bag, rdf:Alt, rdf:_1, rdf:_2 . . . vocabulary to describe(ordered, unordered, alternate) containers of values.rdf:Statement, rdf:subject, rdf:predicate, rdf:object. . . vocabulary to make statements about statement (aka “reification”)rdf:value . . . to describe structured values

Note: The first three are the most important for the formal semntics. Thesemantics doesn’t make any strong restrictions on the grayed out vocabulary.

A. Polleres VU 184.729 5/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Collection vocabulary – Example

Collection vocabulary intended to be used to write lists, e.g.

ex:article ex:hasAuthors ( ex:johnny ex:jim ex:jack )

However, the semantics does not impose strong formal restrictions/conditions on the use ofthis vocabulary, e.g. a graph

ex:a rdf:first ex:a.

is fine (no restrictions on cyclic, incomplete, etc. lists.)

A. Polleres VU 184.729 6/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Collection vocabulary – Example

Collection vocabulary intended to be used to write lists, e.g.

ex:article ex:hasAuthors __ :b1.__ :b1 rdf:first ex:johnny ; rdf:rest __ :b2.__ :b2 rdf:first ex:jim ; rdf:rest __ :b3.__ :b3 rdf:first ex:jack ; rdf:rest rdf:nil.

Comes usually from RDF/XML’s parseType = "Collection".

However, the semantics does not impose strong formal restrictions/conditions on the use ofthis vocabulary, e.g. a graph

ex:a rdf:first ex:a.

is fine (no restrictions on cyclic, incomplete, etc. lists.)

A. Polleres VU 184.729 6/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Collection vocabulary – Example

Collection vocabulary intended to be used to write lists, e.g.

ex:article ex:hasAuthors __ :b1.__ :b1 rdf:first ex:johnny ; rdf:rest __ :b2.__ :b2 rdf:first ex:jim ; rdf:rest __ :b3.__ :b3 rdf:first ex:jack ; rdf:rest rdf:nil.

Comes usually from RDF/XML’s parseType = "Collection".

However, the semantics does not impose strong formal restrictions/conditions on the use ofthis vocabulary, e.g. a graph

ex:a rdf:first ex:a.

is fine (no restrictions on cyclic, incomplete, etc. lists.)

A. Polleres VU 184.729 6/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Container vocabulary – Example

Container vocabulary intended to be used to write containers of values,e.g.

ex:semWebCourse ex:hasParticipants [ a rdf:Bag;rdf:_1 ex:johny,rdf:_2 ex:jim;rdf:_3 ex:jack ] .

ex:semWebCourse ex:hasLecturer [ a rdf:Alt;rdf:_1 ex:prof1rdf:_2 ex:prof2 ] .

Comes usually from RDF/XML’s <li>...</li>.

However, the semantics does not impose strong formal restrictions/conditions on the use ofthis vocabulary, e.g. a graph.

ex:a rdf:_4 rdf:Seq;rdf:_3 rdf:_3 .

is fine (e.g. no restrictions on multiple occurrences of a numbered membership property, etc.).

A. Polleres VU 184.729 7/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Container vocabulary – Example

Container vocabulary intended to be used to write containers of values,e.g.

ex:semWebCourse ex:hasParticipants [ a rdf:Bag;rdf:_1 ex:johny,rdf:_2 ex:jim;rdf:_3 ex:jack ] .

ex:semWebCourse ex:hasLecturer [ a rdf:Alt;rdf:_1 ex:prof1rdf:_2 ex:prof2 ] .

Comes usually from RDF/XML’s <li>...</li>.

However, the semantics does not impose strong formal restrictions/conditions on the use ofthis vocabulary, e.g. a graph.

ex:a rdf:_4 rdf:Seq;rdf:_3 rdf:_3 .

is fine (e.g. no restrictions on multiple occurrences of a numbered membership property, etc.).

A. Polleres VU 184.729 7/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Reification vocabulary & rdf:value – Example

Reification vocab intended to be used to write statements aboutstatements, e.g.

ex:axel ex:thinks [ rdf:subject dbpedia:Web_Ontology_Language;rdf:predicate owl:differentFromrdf:object dbpedia:Description_Logics] .

Again, no semantics restrictions on the use of this vocabulary.

rdf:value intended to be used to denote “structured values”ex:telephone1 ex:hasPrice [ rdf:value 20; ex:currency ‘‘EUR”]

Again, no semantics restrictions on the use of this vocabulary.

Purely speculative statement by the lecturerIntuitively, all these parts of the vocabulary have been left without fixed semantics for themoment. . . may be changed/replaced by later standards? e.g. “Reification-Entailment”, etc.cf. also http://www.w3.org/2011/rdf-wg/track/issues/25

A. Polleres VU 184.729 8/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Reification vocabulary & rdf:value – Example

Reification vocab intended to be used to write statements aboutstatements, e.g.

ex:axel ex:thinks [ rdf:subject dbpedia:Web_Ontology_Language;rdf:predicate owl:differentFromrdf:object dbpedia:Description_Logics] .

Again, no semantics restrictions on the use of this vocabulary.

rdf:value intended to be used to denote “structured values”ex:telephone1 ex:hasPrice [ rdf:value 20; ex:currency ‘‘EUR”]

Again, no semantics restrictions on the use of this vocabulary.

Purely speculative statement by the lecturerIntuitively, all these parts of the vocabulary have been left without fixed semantics for themoment. . . may be changed/replaced by later standards? e.g. “Reification-Entailment”, etc.cf. also http://www.w3.org/2011/rdf-wg/track/issues/25

A. Polleres VU 184.729 8/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Reification vocabulary & rdf:value – Example

Reification vocab intended to be used to write statements aboutstatements, e.g.

ex:axel ex:thinks [ rdf:subject dbpedia:Web_Ontology_Language;rdf:predicate owl:differentFromrdf:object dbpedia:Description_Logics] .

Again, no semantics restrictions on the use of this vocabulary.

rdf:value intended to be used to denote “structured values”ex:telephone1 ex:hasPrice [ rdf:value 20; ex:currency ‘‘EUR”]

Again, no semantics restrictions on the use of this vocabulary.

Purely speculative statement by the lecturerIntuitively, all these parts of the vocabulary have been left without fixed semantics for themoment. . . may be changed/replaced by later standards? e.g. “Reification-Entailment”, etc.cf. also http://www.w3.org/2011/rdf-wg/track/issues/25

A. Polleres VU 184.729 8/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

. . . now to the formal part.

Recall from last time (lecture 2, slide 24):

Simple Interpretation

A simple interpretation I over vocabulary V is a 6-tupleI = 〈IR, IP, IEXT, IS, IL, LV 〉

A. Polleres VU 184.729 9/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-interpretation

An rdf-interpretation is a simple interpretationI = 〈IR, IP, IEXT, IS, IL, LV 〉 which fulfills the following conditions and allthe folllowing set of axiomatic triples Ardf :

rdf-satisaction, entailmentwritten |=rdf defined analogously to |=, but with respect to rdf-interpretations.

A. Polleres VU 184.729 10/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-interpretation

An rdf-interpretation is a simple interpretationI = 〈IR, IP, IEXT, IS, IL, LV 〉 which fulfills the following conditions and allthe folllowing set of axiomatic triples Ardf :

rdf-satisaction, entailmentwritten |=rdf defined analogously to |=, but with respect to rdf-interpretations.

A. Polleres VU 184.729 10/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-interpretation

Some consequences:

Proposition 1If s p o. ∈ G then G |=rdf {p a rdf:Property}.

Particularly, note that the first axiomatic triple is redundant!

Let LXML denote the set of well-formed XML literals.

Proposition 2If s p "xyz"∧∧rdf:XMLLiteral. ∈ G, such that xyz ∈ LXML thenG |=rdf {[] a rdf:XMLLiteral}.

Remark: ill-formed XML literals are hardly (impossibly?) writeable in RDF/XML, but in other

syntaxes.

A. Polleres VU 184.729 11/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-interpretation

Some consequences:

Proposition 1If s p o. ∈ G then G |=rdf {p a rdf:Property}.

Particularly, note that the first axiomatic triple is redundant!

Let LXML denote the set of well-formed XML literals.

Proposition 2If s p "xyz"∧∧rdf:XMLLiteral. ∈ G, such that xyz ∈ LXML thenG |=rdf {[] a rdf:XMLLiteral}.

Remark: ill-formed XML literals are hardly (impossibly?) writeable in RDF/XML, but in other

syntaxes.

A. Polleres VU 184.729 11/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-interpretation

Some consequences:

Proposition 1If s p o. ∈ G then G |=rdf {p a rdf:Property}.

Particularly, note that the first axiomatic triple is redundant!

Let LXML denote the set of well-formed XML literals.

Proposition 2If s p "xyz"∧∧rdf:XMLLiteral. ∈ G, such that xyz ∈ LXML thenG |=rdf {[] a rdf:XMLLiteral}.

Remark: ill-formed XML literals are hardly (impossibly?) writeable in RDF/XML, but in other

syntaxes.

A. Polleres VU 184.729 11/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-interpretation

Some consequences:

Proposition 1If s p o. ∈ G then G |=rdf {p a rdf:Property}.

Particularly, note that the first axiomatic triple is redundant!

Let LXML denote the set of well-formed XML literals.

Proposition 2If s p "xyz"∧∧rdf:XMLLiteral. ∈ G, such that xyz ∈ LXML thenG |=rdf {[] a rdf:XMLLiteral}.

Remark: ill-formed XML literals are hardly (impossibly?) writeable in RDF/XML, but in other

syntaxes.

A. Polleres VU 184.729 11/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Computing entailment for |=rdf

Recall main result for simple entailment (|=):

Interpolation LemmaS |= E if and only if a subgraph of S is an instance of E.

Does no longer hold for |=rdf :-(

BUT! Can be enabled by rules!

rdf-closure ( Clrdf (G) )

The rdf-closure Clrdf (G) of graph G is defined by G ]Ardf plus exhaustiveapplication of the following rules to G:

A. Polleres VU 184.729 12/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Computing entailment for |=rdf

Recall main result for simple entailment (|=):

Interpolation LemmaS |= E if and only if a subgraph of S is an instance of E.

Does no longer hold for |=rdf :-(

BUT! Can be enabled by rules!

rdf-closure ( Clrdf (G) )

The rdf-closure Clrdf (G) of graph G is defined by G ]Ardf plus exhaustiveapplication of the following rules to G:

If s p o.∈ Clrdf (G) then add p a rdf:Property.

If s p "xyz"∧∧rdf:XMLLiteral.∈ Clrdf (G) and xyz ∈ LXML then add __ :xyz ardf:XMLLiteral. where __ :xyz is a distinct bnode for that literal.

Slight abstraction of derivation rules rdf1 and rdf2 from the spec.

A. Polleres VU 184.729 12/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Computing entailment for |=rdf

Recall main result for simple entailment (|=):

Interpolation LemmaS |= E if and only if a subgraph of S is an instance of E.

Does no longer hold for |=rdf :-(

BUT! Can be enabled by rules!

rdf-closure ( Clrdf (G) )

The rdf-closure Clrdf (G) of graph G is defined by G ]Ardf plus exhaustiveapplication of the following rules to G:

p a rdf:Property . ⇐ s p o .

__ :xyz a rdf:XMLLiteral . ⇐ s p "xyz"∧∧rdf:XMLLiteral.

Let’s write them in more “rule” syntax.

A. Polleres VU 184.729 12/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Computing entailment for |=rdf

Recall main result for simple entailment (|=):

Interpolation LemmaS |= E if and only if a subgraph of S is an instance of E.

Does no longer hold for |=rdf :-(

BUT! Can be enabled by rules!

rdf-closure ( Clrdf (G) )

The rdf-closure Clrdf (G) of graph G is defined by G ]Ardf plus exhaustiveapplication of the following rules to G:

∀s, p, o(triple(p, rdf:type, rdf:Property)← triple(s, p, o))

∀s, p∃b(triple(b, rdf:type, rdf:XMLLiteral)← triple(s, p, "xyz"∧∧rdf:XMLLiteral))

Or as first-order sentences.

A. Polleres VU 184.729 12/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

Computing entailment for |=rdf

Recall main result for simple entailment (|=):

Interpolation LemmaS |= E if and only if a subgraph of S is an instance of E.

Does no longer hold for |=rdf :-(

BUT! Can be enabled by rules!

rdf-closure ( Clrdf (G) )

The rdf-closure Clrdf (G) of graph G is defined by G ]Ardf plus exhaustiveapplication of the following rules to G:

triple(P,rdf:type,rdf:Property) :- triple(S,P,O).

triple(bxyz,rdf:type,rdf:XMLLiteral) :- triple(S,P,"xyz"∧∧rdf:XMLLiteral)

Or as Datalog rules.

A. Polleres VU 184.729 12/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-entailment: Properties

RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S) |= E

Wait, slight problem for actual computation. . .Clrdf (G) is infinite because of Ardf .

But: A finite subset of Clrdf (G) is sufficient! Let the reduced closure of G1 w.r.t. G2, writtenClrdf (G1, G2), be defined as Clrdf (G1) with only those axiomatic triples rdf:_n rdf:typerdf:Property . where rdf:_n∈ V (G2)

⇒ For a finite graph G2, Clrdf (G1, G2) is always finite.

Modified RDF entailment lemma (slightly adapted from the spec)S |=rdf E if and only if Clrdf (S,E) |= E

Means that we can compute RDF entailment finitely! :-)

A. Polleres VU 184.729 13/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-entailment: Properties

RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S) |= E

Wait, slight problem for actual computation. . .Clrdf (G) is infinite because of Ardf .

But: A finite subset of Clrdf (G) is sufficient! Let the reduced closure of G1 w.r.t. G2, writtenClrdf (G1, G2), be defined as Clrdf (G1) with only those axiomatic triples rdf:_n rdf:typerdf:Property . where rdf:_n∈ V (G2)

⇒ For a finite graph G2, Clrdf (G1, G2) is always finite.

Modified RDF entailment lemma (slightly adapted from the spec)S |=rdf E if and only if Clrdf (S,E) |= E

Means that we can compute RDF entailment finitely! :-)

A. Polleres VU 184.729 13/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-entailment: Properties

RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S) |= E

Wait, slight problem for actual computation. . .Clrdf (G) is infinite because of Ardf .

But: A finite subset of Clrdf (G) is sufficient! Let the reduced closure of G1 w.r.t. G2, writtenClrdf (G1, G2), be defined as Clrdf (G1) with only those axiomatic triples rdf:_n rdf:typerdf:Property . where rdf:_n∈ V (G2)

⇒ For a finite graph G2, Clrdf (G1, G2) is always finite.

Modified RDF entailment lemma (slightly adapted from the spec)S |=rdf E if and only if Clrdf (S,E) |= E

Means that we can compute RDF entailment finitely! :-)

A. Polleres VU 184.729 13/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-entailment: Properties

RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S) |= E

Wait, slight problem for actual computation. . .Clrdf (G) is infinite because of Ardf .

But: A finite subset of Clrdf (G) is sufficient! Let the reduced closure of G1 w.r.t. G2, writtenClrdf (G1, G2), be defined as Clrdf (G1) with only those axiomatic triples rdf:_n rdf:typerdf:Property . where rdf:_n∈ V (G2)

⇒ For a finite graph G2, Clrdf (G1, G2) is always finite.

Modified RDF entailment lemma (slightly adapted from the spec)S |=rdf E if and only if Clrdf (S,E) |= E

Means that we can compute RDF entailment finitely! :-)

A. Polleres VU 184.729 13/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-entailment: Properties

RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S) |= E

Wait, slight problem for actual computation. . .Clrdf (G) is infinite because of Ardf .

But: A finite subset of Clrdf (G) is sufficient! Let the reduced closure of G1 w.r.t. G2, writtenClrdf (G1, G2), be defined as Clrdf (G1) with only those axiomatic triples rdf:_n rdf:typerdf:Property . where rdf:_n∈ V (G2)

⇒ For a finite graph G2, Clrdf (G1, G2) is always finite.

Modified RDF entailment lemma (slightly adapted from the spec)S |=rdf E if and only if Clrdf (S,E) |= E

Means that we can compute RDF entailment finitely! :-)

A. Polleres VU 184.729 13/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

rdf-entailment: Properties

RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S) |= E

Wait, slight problem for actual computation. . .Clrdf (G) is infinite because of Ardf .

But: A finite subset of Clrdf (G) is sufficient! Let the reduced closure of G1 w.r.t. G2, writtenClrdf (G1, G2), be defined as Clrdf (G1) with only those axiomatic triples rdf:_n rdf:typerdf:Property . where rdf:_n∈ V (G2)

⇒ For a finite graph G2, Clrdf (G1, G2) is always finite.

Modified RDF entailment lemma (slightly adapted from the spec)S |=rdf E if and only if Clrdf (S,E) |= E

Means that we can compute RDF entailment finitely! :-)

A. Polleres VU 184.729 13/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

So what?

Now that was still fairly uninteresting. . .

. . . Didn’t buy us too much, except that:entailment became more complicatedwe have a way to express that a predicate is an rdf:Propery

we have a way to express that XML Literals are rdf:XMLLiterals

Still, we cannot describe other vocabularies ⇒ rdfs:

A. Polleres VU 184.729 14/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

So what?

Now that was still fairly uninteresting. . .

. . . Didn’t buy us too much, except that:entailment became more complicatedwe have a way to express that a predicate is an rdf:Propery

we have a way to express that XML Literals are rdf:XMLLiterals

Still, we cannot describe other vocabularies ⇒ rdfs:

A. Polleres VU 184.729 14/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.1 RDF Entailment

So what?

Now that was still fairly uninteresting. . .

. . . Didn’t buy us too much, except that:entailment became more complicatedwe have a way to express that a predicate is an rdf:Propery

we have a way to express that XML Literals are rdf:XMLLiterals

Still, we cannot describe other vocabularies ⇒ rdfs:

A. Polleres VU 184.729 14/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

The rdfs: vocabulary

The RDFS vocabulary, rdfsV, extends rdfV:rdfs:domain, rdfs:range

rdfs:subClassOf, rdfs:subPropertyOf

rdfs:Resource . . . all resources.

rdfs:Class . . . all resources which denote classes.

rdfs:Literal . . . the “class of literal values”.

rdfs:Datatype . . . the “class of datatypes”.

rdfs:Container . . . a joint superclass of rdf:Alt, rdf:Seq, rdf:Bag.

rdfs:ContainerMembershipProperty . . . the “class of conainer membership predicates”(rdf:_1, rdf:_2,. . . ).

rdfs:member a joint superproperty of rdf:_1, rdf:_2,. . . .

rdfs:comment, rdfs:seeAlso, rdfs:isDefinedBy, rdfs:label some moreconventional vocabulary without strong “formal” semantics1

1for illustrative examples, let’s check the FOAF ontologyRDF/XML version: http://xmlns.com/foaf/spec/20071002.rdfExtracted Turtle version:http://www.polleres.net/teaching/SemWebTech_2009/testdata/foafOntology.ttl

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretation

rdfs-interpretation

An rdfs-interpretation is an rdf-interpretation I defined by additional

semantic conditions

axiomatic triples Ardfs

Additionally to IR and IP , the semantic conditions for RDF also talkabout the set of classes, denoted by IC ⊆ IR:

ICIC is defined as the set of resources which appear in the range of the extension of rdf:type,i.e. all the y such that 〈x, y〉 ∈ IEXT (I(rdf:type)).

For brevity, we also define:

ICEXTICEXT : IC → 2IR defined as x ∈ ICEXT (y)⇔ 〈x, y〉 ∈ IEXT (I(rdf:type))

Let’s look into the semantic conditions and axiomatic triples for rdfs-interpretations now. . .

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretation

rdfs-interpretation

An rdfs-interpretation is an rdf-interpretation I defined by additional

semantic conditions

axiomatic triples Ardfs

Additionally to IR and IP , the semantic conditions for RDF also talkabout the set of classes, denoted by IC ⊆ IR:

ICIC is defined as the set of resources which appear in the range of the extension of rdf:type,i.e. all the y such that 〈x, y〉 ∈ IEXT (I(rdf:type)).

For brevity, we also define:

ICEXTICEXT : IC → 2IR defined as x ∈ ICEXT (y)⇔ 〈x, y〉 ∈ IEXT (I(rdf:type))

Let’s look into the semantic conditions and axiomatic triples for rdfs-interpretations now. . .

A. Polleres VU 184.729 16/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretation

rdfs-interpretation

An rdfs-interpretation is an rdf-interpretation I defined by additional

semantic conditions

axiomatic triples Ardfs

Additionally to IR and IP , the semantic conditions for RDF also talkabout the set of classes, denoted by IC ⊆ IR:

ICIC is defined as the set of resources which appear in the range of the extension of rdf:type,i.e. all the y such that 〈x, y〉 ∈ IEXT (I(rdf:type)).

For brevity, we also define:

ICEXTICEXT : IC → 2IR defined as x ∈ ICEXT (y)⇔ 〈x, y〉 ∈ IEXT (I(rdf:type))

Let’s look into the semantic conditions and axiomatic triples for rdfs-interpretations now. . .

A. Polleres VU 184.729 16/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretation

rdfs-interpretation

An rdfs-interpretation is an rdf-interpretation I defined by additional

semantic conditions

axiomatic triples Ardfs

Additionally to IR and IP , the semantic conditions for RDF also talkabout the set of classes, denoted by IC ⊆ IR:

ICIC is defined as the set of resources which appear in the range of the extension of rdf:type,i.e. all the y such that 〈x, y〉 ∈ IEXT (I(rdf:type)).

For brevity, we also define:

ICEXTICEXT : IC → 2IR defined as x ∈ ICEXT (y)⇔ 〈x, y〉 ∈ IEXT (I(rdf:type))

Let’s look into the semantic conditions and axiomatic triples for rdfs-interpretations now. . .

A. Polleres VU 184.729 16/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretation

rdfs-interpretation

An rdfs-interpretation is an rdf-interpretation I defined by additional

semantic conditions

axiomatic triples Ardfs

Additionally to IR and IP , the semantic conditions for RDF also talkabout the set of classes, denoted by IC ⊆ IR:

ICIC is defined as the set of resources which appear in the range of the extension of rdf:type,i.e. all the y such that 〈x, y〉 ∈ IEXT (I(rdf:type)).

For brevity, we also define:

ICEXTICEXT : IC → 2IR defined as x ∈ ICEXT (y)⇔ 〈x, y〉 ∈ IEXT (I(rdf:type))

Let’s look into the semantic conditions and axiomatic triples for rdfs-interpretations now. . .

A. Polleres VU 184.729 16/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Semantic conditions

o a Class . ⇐ s a o .[] a s . ⇐ s a Class .s a Resource . ⇐ s p o .p a Resource . ⇐ s p o .o a Resource . ⇐ s p o . o 6∈ L__ :l a Literal . ⇐ s p l. l ∈ Ls p __ :l . ⇐ s p l . l ∈ L

s a c . ⇐ p domain c . s p o .

o a c . ⇐ p range c . s p o . o 6∈ L

s subPropertyOf r . ⇐ s subPropertyOf o .o subPropertyOf r .

s subPropertyOf s . ⇐ s a Property .s a Property . ⇐ s subPropertyOf o .o a Property . ⇐ s subPropertyOf o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BLs subClassOf Resource . ⇐ s a Class .s a Class . ⇐ s subClassOf o .o a Class . ⇐ s subClassOf o . o 6∈ Ls a c . ⇐ o suClassOf c . s a o .s subClassOf r . ⇐ s subClassOf o .

o subClassOf r .s subClassOf s . ⇐ s a Property .s subPropertyOf member .⇐

s a ContainerMembershipProperty.

s subClassOf Literal . ⇐ s a Datatype.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Semantic conditionso a Class . ⇐ s a o .[] a s . ⇐ s a Class .s a Resource . ⇐ s p o .p a Resource . ⇐ s p o .o a Resource . ⇐ s p o . o 6∈ L__ :l a Literal . ⇐ s p l. l ∈ Ls p __ :l . ⇐ s p l . l ∈ L

s a c . ⇐ p domain c . s p o .

o a c . ⇐ p range c . s p o . o 6∈ L

s subPropertyOf r . ⇐ s subPropertyOf o .o subPropertyOf r .

s subPropertyOf s . ⇐ s a Property .s a Property . ⇐ s subPropertyOf o .o a Property . ⇐ s subPropertyOf o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BLs subClassOf Resource . ⇐ s a Class .s a Class . ⇐ s subClassOf o .o a Class . ⇐ s subClassOf o . o 6∈ Ls a c . ⇐ o suClassOf c . s a o .s subClassOf r . ⇐ s subClassOf o .

o subClassOf r .s subClassOf s . ⇐ s a Property .s subPropertyOf member .⇐

s a ContainerMembershipProperty.

s subClassOf Literal . ⇐ s a Datatype.

A. Polleres VU 184.729 17/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Semantic conditionso a Class . ⇐ s a o .[] a s . ⇐ s a Class .s a Resource . ⇐ s p o .p a Resource . ⇐ s p o .o a Resource . ⇐ s p o . o 6∈ L__ :l a Literal . ⇐ s p l. l ∈ Ls p __ :l . ⇐ s p l . l ∈ L

s a c . ⇐ p domain c . s p o .

o a c . ⇐ p range c . s p o . o 6∈ L

s subPropertyOf r . ⇐ s subPropertyOf o .o subPropertyOf r .

s subPropertyOf s . ⇐ s a Property .s a Property . ⇐ s subPropertyOf o .o a Property . ⇐ s subPropertyOf o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BLs subClassOf Resource . ⇐ s a Class .s a Class . ⇐ s subClassOf o .o a Class . ⇐ s subClassOf o . o 6∈ Ls a c . ⇐ o suClassOf c . s a o .s subClassOf r . ⇐ s subClassOf o .

o subClassOf r .s subClassOf s . ⇐ s a Property .s subPropertyOf member .⇐

s a ContainerMembershipProperty.

s subClassOf Literal . ⇐ s a Datatype.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Semantic conditionso a Class . ⇐ s a o .[] a s . ⇐ s a Class .s a Resource . ⇐ s p o .p a Resource . ⇐ s p o .o a Resource . ⇐ s p o . o 6∈ L__ :l a Literal . ⇐ s p l. l ∈ Ls p __ :l . ⇐ s p l . l ∈ L

s a c . ⇐ p domain c . s p o .

o a c . ⇐ p range c . s p o . o 6∈ L

s subPropertyOf r . ⇐ s subPropertyOf o .o subPropertyOf r .

s subPropertyOf s . ⇐ s a Property .s a Property . ⇐ s subPropertyOf o .o a Property . ⇐ s subPropertyOf o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BLs subClassOf Resource . ⇐ s a Class .s a Class . ⇐ s subClassOf o .o a Class . ⇐ s subClassOf o . o 6∈ Ls a c . ⇐ o suClassOf c . s a o .s subClassOf r . ⇐ s subClassOf o .

o subClassOf r .s subClassOf s . ⇐ s a Property .s subPropertyOf member .⇐

s a ContainerMembershipProperty.

s subClassOf Literal . ⇐ s a Datatype.

A. Polleres VU 184.729 17/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Semantic conditionso a Class . ⇐ s a o .[] a s . ⇐ s a Class .s a Resource . ⇐ s p o .p a Resource . ⇐ s p o .o a Resource . ⇐ s p o . o 6∈ L__ :l a Literal . ⇐ s p l. l ∈ Ls p __ :l . ⇐ s p l . l ∈ L

s a c . ⇐ p domain c . s p o .

o a c . ⇐ p range c . s p o . o 6∈ L

s subPropertyOf r . ⇐ s subPropertyOf o .o subPropertyOf r .

s subPropertyOf s . ⇐ s a Property .s a Property . ⇐ s subPropertyOf o .o a Property . ⇐ s subPropertyOf o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BLs subClassOf Resource . ⇐ s a Class .s a Class . ⇐ s subClassOf o .o a Class . ⇐ s subClassOf o . o 6∈ Ls a c . ⇐ o suClassOf c . s a o .s subClassOf r . ⇐ s subClassOf o .

o subClassOf r .s subClassOf s . ⇐ s a Property .s subPropertyOf member .⇐

s a ContainerMembershipProperty.

s subClassOf Literal . ⇐ s a Datatype.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Additional Axiomatic TriplesArdfs

rdf:type rdfs:domain rdfs:Resource;rdfs:range rdfs:Class .

rdfs:domain rdfs:domain rdf:Property;rdfs:range rdfs:Class .

rdfs:range rdfs:domain rdf:Property;rdfs:range rdfs:Class .

rdfs:subPropertyOf rdfs:domain rdf:Property;rdfs:range rdf:Property .

rdfs:subClassOf rdfs:domain rdfs:Class;rdfs:range rdfs:Class .

rdfs:member rdfs:domain rdfs:Resource;rdfs:range rdfs:Resource .

rdfs:seeAlso rdfs:domain rdfs:Resource;rdfs:range rdfs:Resource .

rdfs:isDefinedBy rdfs:domain rdfs:Resource;rdfs:range rdfs:Resource .

rdf:value rdfs:domain rdfs:Resource;rdfs:range rdfs:Resource .

rdfs:comment rdfs:domain rdfs:Resource;rdfs:range rdfs:Literal .

rdfs:label rdfs:domain rdfs:Resource;rdfs:range rdfs:Literal .

rdf:first rdfs:domain rdf:List;rdfs:range rdfs:Resource .

rdf:rest rdfs:domain rdf:List .rdfs:range rdf:List .

rdf:subject rdfs:domain rdf:Statement;rdfs:range rdfs:Resource .

rdf:predicate rdfs:domain rdf:Statement;rdfs:range rdfs:Resource .

rdf:object rdfs:domain rdf:Statement;rdfs:range rdfs:Resource .

rdf:Alt rdfs:subClassOf rdfs:Container .rdf:Bag rdfs:subClassOf rdfs:Container .rdf:Seq rdfs:subClassOf rdfs:Container .rdfs:ContainerMembershipProperty

rdfs:subClassOf rdf:Property .

rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .

rdf:XMLLiteral rdf:type rdfs:Datatype .rdf:XMLLiteral rdfs:subClassOf rdfs:Literal .rdfs:Datatype rdfs:subClassOf rdfs:Class .

rdf:_1 rdf:type rdfs:ContainerMembershipProperty .rdf:_1 rdfs:domain rdfs:Resource .rdf:_1 rdfs:range rdfs:Resource .rdf:_2 rdf:type rdfs:ContainerMembershipProperty .rdf:_2 rdfs:domain rdfs:Resource .rdf:_2 rdfs:range rdfs:Resource ....

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Axiomatic Triples

Ardfs has some “somewhat redundant triples” ( let’s call the “gist” A−rdfs)

rdf:type rdfs:range rdfs:Class .rdfs:domain rdfs:range rdfs:Class .rdfs:range rdfs:range rdfs:Class .

rdfs:subClassOf rdfs:domain rdfs:Class;rdfs:range rdfs:Class .

rdfs:comment rdfs:range rdfs:Literal .rdfs:label rdfs:range rdfs:Literal .

rdf:first rdfs:domain rdf:List .rdf:rest rdfs:domain rdf:List .

rdfs:range rdf:List .

rdf:subject rdfs:domain rdf:Statement;rdf:predicate rdfs:domain rdf:Statement;rdf:object rdfs:domain rdf:Statement;

rdf:Alt rdfs:subClassOf rdfs:Container .rdf:Bag rdfs:subClassOf rdfs:Container .rdf:Seq rdfs:subClassOf rdfs:Container .rdfs:ContainerMembershipProperty

rdfs:subClassOf rdf:Property .

rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .

rdf:XMLLiteral rdf:type rdfs:Datatype .rdf:XMLLiteral rdfs:subClassOf rdfs:Literal .rdfs:Datatype rdfs:subClassOf rdfs:Class .

rdf:_1 rdf:type rdfs:ContainerMembershipProperty .rdf:_2 rdf:type rdfs:ContainerMembershipProperty ....

Actually, one could think: A−rdfs is enough:

Axiomatic triples s domain/range rdfs:Resource .Axiomatic triples p domain/range rdf:Property .

More on that later. . . ... bottomline: if we aren’t interested in inferring triples

x rdfs:domain rdfs:Resource. , we can probably do without these.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-interpretations: Axiomatic Triples

Ardfs has some “somewhat redundant triples” ( let’s call the “gist” A−rdfs)

rdf:type rdfs:range rdfs:Class .rdfs:domain rdfs:range rdfs:Class .rdfs:range rdfs:range rdfs:Class .

rdfs:subClassOf rdfs:domain rdfs:Class;rdfs:range rdfs:Class .

rdfs:comment rdfs:range rdfs:Literal .rdfs:label rdfs:range rdfs:Literal .

rdf:first rdfs:domain rdf:List .rdf:rest rdfs:domain rdf:List .

rdfs:range rdf:List .

rdf:subject rdfs:domain rdf:Statement;rdf:predicate rdfs:domain rdf:Statement;rdf:object rdfs:domain rdf:Statement;

rdf:Alt rdfs:subClassOf rdfs:Container .rdf:Bag rdfs:subClassOf rdfs:Container .rdf:Seq rdfs:subClassOf rdfs:Container .rdfs:ContainerMembershipProperty

rdfs:subClassOf rdf:Property .

rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .

rdf:XMLLiteral rdf:type rdfs:Datatype .rdf:XMLLiteral rdfs:subClassOf rdfs:Literal .rdfs:Datatype rdfs:subClassOf rdfs:Class .

rdf:_1 rdf:type rdfs:ContainerMembershipProperty .rdf:_2 rdf:type rdfs:ContainerMembershipProperty ....

Actually, one could think: A−rdfs is enough:

Axiomatic triples s domain/range rdfs:Resource .Axiomatic triples p domain/range rdf:Property .

More on that later. . . ... bottomline: if we aren’t interested in inferring triples

x rdfs:domain rdfs:Resource. , we can probably do without these.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Properties

Recall:

Modified RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S,E) |= E

Can we do something similar for rdfs?

Good news: Yes! Extension from Clrdf (G,E) to Clrdfs(G,E) is straightforward, just add thenew inference rules.

RDF Spec[Hayes, 2004, Section 7] provides an informative set of entailment rules to captureRDF- and RDFS entailment. Pretty much our rules in the last slide.

Some errors found later, several refinements in the literature.[ter Horst, 2005][Muñoz et al.,2007][Bruijn and Heymans, 2007][Ianni et al., 2009].

. . . but, means that we can compute RDFS entailment finitely! :-)

A. Polleres VU 184.729 20/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Properties

Recall:

Modified RDF entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdf (S,E) |= E

Can we do something similar for rdfs?

Good news: Yes! Extension from Clrdf (G,E) to Clrdfs(G,E) is straightforward, just add thenew inference rules.

RDF Spec[Hayes, 2004, Section 7] provides an informative set of entailment rules to captureRDF- and RDFS entailment. Pretty much our rules in the last slide.

Some errors found later, several refinements in the literature.[ter Horst, 2005][Muñoz et al.,2007][Bruijn and Heymans, 2007][Ianni et al., 2009].

. . . but, means that we can compute RDFS entailment finitely! :-)

A. Polleres VU 184.729 20/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Properties

Recall:

Modified RDFS entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdfs(S,E) |= E

Can we do something similar for rdfs?

Good news: Yes! Extension from Clrdf (G,E) to Clrdfs(G,E) is straightforward, just add thenew inference rules.

RDF Spec[Hayes, 2004, Section 7] provides an informative set of entailment rules to captureRDF- and RDFS entailment. Pretty much our rules in the last slide.

Some errors found later, several refinements in the literature.[ter Horst, 2005][Muñoz et al.,2007][Bruijn and Heymans, 2007][Ianni et al., 2009].

. . . but, means that we can compute RDFS entailment finitely! :-)

A. Polleres VU 184.729 20/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Properties

Recall:

Modified RDFS entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdfs(S,E) |= E

Can we do something similar for rdfs?

Good news: Yes! Extension from Clrdf (G,E) to Clrdfs(G,E) is straightforward, just add thenew inference rules.

RDF Spec[Hayes, 2004, Section 7] provides an informative set of entailment rules to captureRDF- and RDFS entailment. Pretty much our rules in the last slide.

Some errors found later, several refinements in the literature.[ter Horst, 2005][Muñoz et al.,2007][Bruijn and Heymans, 2007][Ianni et al., 2009].

. . . but, means that we can compute RDFS entailment finitely! :-)

A. Polleres VU 184.729 20/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Properties

Recall:

Modified RDFS entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdfs(S,E) |= E

Can we do something similar for rdfs?

Good news: Yes! Extension from Clrdf (G,E) to Clrdfs(G,E) is straightforward, just add thenew inference rules.

RDF Spec[Hayes, 2004, Section 7] provides an informative set of entailment rules to captureRDF- and RDFS entailment. Pretty much our rules in the last slide.

Some errors found later, several refinements in the literature.[ter Horst, 2005][Muñoz et al.,2007][Bruijn and Heymans, 2007][Ianni et al., 2009].

. . . but, means that we can compute RDFS entailment finitely! :-)

A. Polleres VU 184.729 20/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Properties

Recall:

Modified RDFS entailment lemma (slightly adapted from the spec)

S |=rdf E if and only if Clrdfs(S,E) |= E

Can we do something similar for rdfs?

Good news: Yes! Extension from Clrdf (G,E) to Clrdfs(G,E) is straightforward, just add thenew inference rules.

RDF Spec[Hayes, 2004, Section 7] provides an informative set of entailment rules to captureRDF- and RDFS entailment. Pretty much our rules in the last slide.

Some errors found later, several refinements in the literature.[ter Horst, 2005][Muñoz et al.,2007][Bruijn and Heymans, 2007][Ianni et al., 2009].

. . . but, means that we can compute RDFS entailment finitely! :-)

A. Polleres VU 184.729 20/43

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Spec entailment rules are – incomplete!Some of the rules we had before had restrictions in the antecedent, e.g.

s a c . ⇐ p domain c . s p o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BL

But what about that:

p rdfs:subPropertyOf _:p_:p rdfs:domain c.s p o .

Problem: If we execute the rules with the restrictions in place, we won’t get:

s rdf:type c.

The invalid “intermediate” triple

s _:p c.

cannot be inferred “within RDF”.

Solution: easy, (i) allow “generalized” RDF (UBL× UBL× UBL ) for intermediatecomputation of closure, or (ii) use modified “implicit typing rule”, cf. [Muñoz et al., 2007]:

s a c ⇐ p subPropertyOf q. q rdfs:domain c. s p o.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Spec entailment rules are – incomplete!Some of the rules we had before had restrictions in the antecedent, e.g.

s a c . ⇐ p domain c . s p o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BL

But what about that:

p rdfs:subPropertyOf _:p_:p rdfs:domain c.s p o .

Problem: If we execute the rules with the restrictions in place, we won’t get:

s rdf:type c.

The invalid “intermediate” triple

s _:p c.

cannot be inferred “within RDF”.

Solution: easy, (i) allow “generalized” RDF (UBL× UBL× UBL ) for intermediatecomputation of closure, or (ii) use modified “implicit typing rule”, cf. [Muñoz et al., 2007]:

s a c ⇐ p subPropertyOf q. q rdfs:domain c. s p o.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Spec entailment rules are – incomplete!Some of the rules we had before had restrictions in the antecedent, e.g.

s a c . ⇐ p domain c . s p o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BL

But what about that:

p rdfs:subPropertyOf _:p_:p rdfs:domain c.s p o .

Problem: If we execute the rules with the restrictions in place, we won’t get:

s rdf:type c.

The invalid “intermediate” triple

s _:p c.

cannot be inferred “within RDF”.

Solution: easy, (i) allow “generalized” RDF (UBL× UBL× UBL ) for intermediatecomputation of closure, or (ii) use modified “implicit typing rule”, cf. [Muñoz et al., 2007]:

s a c ⇐ p subPropertyOf q. q rdfs:domain c. s p o.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Spec entailment rules are – incomplete!Some of the rules we had before had restrictions in the antecedent, e.g.

s a c . ⇐ p domain c . s p o .s q o ⇐ p subPropertyOf q. s p o. q 6∈ BL

But what about that:

p rdfs:subPropertyOf _:p_:p rdfs:domain c.s p o .

Problem: If we execute the rules with the restrictions in place, we won’t get:

s rdf:type c.

The invalid “intermediate” triple

s _:p c.

cannot be inferred “within RDF”.

Solution: easy, (i) allow “generalized” RDF (UBL× UBL× UBL ) for intermediatecomputation of closure, or (ii) use modified “implicit typing rule”, cf. [Muñoz et al., 2007]:

s a c ⇐ p subPropertyOf q. q rdfs:domain c. s p o.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Inconsistency

Proposition 3ANY Graph

containing a triple s p "xyz"∧∧rdf:XMLLiteral. such that xyz is NOT a well-formedXML literal and

allowing to infer the triple p rdf:range rdf:XMLLiteral .

is false in any rdfs-interpretation.

Why?

by semantic conditions of rdf- and rdfs-inerpretations!

Implication:

There can be rdfs-inconsistent RDF graphs, e.g.

ex:a ex:p "<notLegalXML"^^rdf:XMLLiteral .ex:p rdfs:range rdf:XMLLiteral .

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Inconsistency

Proposition 3ANY Graph

containing a triple s p "xyz"∧∧rdf:XMLLiteral. such that xyz is NOT a well-formedXML literal and

allowing to infer the triple p rdf:range rdf:XMLLiteral .

is false in any rdfs-interpretation.

Why?

by semantic conditions of rdf- and rdfs-inerpretations!

Implication:

There can be rdfs-inconsistent RDF graphs, e.g.

ex:a ex:p "<notLegalXML"^^rdf:XMLLiteral .ex:p rdfs:range rdf:XMLLiteral .

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

Inconsistency

Proposition 3ANY Graph

containing a triple s p "xyz"∧∧rdf:XMLLiteral. such that xyz is NOT a well-formedXML literal and

allowing to infer the triple p rdf:range rdf:XMLLiteral .

is false in any rdfs-interpretation.

Why?

by semantic conditions of rdf- and rdfs-inerpretations!

Implication:

There can be rdfs-inconsistent RDF graphs, e.g.

ex:a ex:p "<notLegalXML"^^rdf:XMLLiteral .ex:p rdfs:range rdf:XMLLiteral .

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Example 1

{} |=rdfs G5

G5 :

rdfs:Resource rdf:type rdfs:Class.rdf:Property rdf:type rdfs:Resource.rdf:Property rdf:subclassOf rdfs:Resource.rdf:Property rdf:type rdfs:Class.rdfs:Class rdf:type rdfs:Resource.rdfs:Class rdf:type rdfs:Class.rdfs:Class rdf:subclassOf rdfs:Resource.rdfs:Class rdf:subclassOf rdfs:Class.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Example 1

{} |=rdfs G5

G5 :

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Example 2

G1 |=rdfs G2

G1 :

ex:alice foaf:knows ex:bob.ex:alice foaf:name "Alice".foaf:knows rdfs:domain foaf:Person.

G2 :

ex:alice rdf:type foaf:Person.

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Unit 3 – RDF(S) & OWL Semantics 1. RDF(S) Entailment 1.2 RDF Entailment

rdfs-entailment: Example 3

{} |=rdfs { xxx rdf:type rdfs:Resource. }

for all xxx in the Vocabulary of any interpretation I.

Implication: Since all rdfs-Interpretations are over an infinite vocabulary(rdf:_n ), even without the axiomatic triples that would already entail infinitetriples.

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

Unit Outline

1. RDF(S) Entailment: Giving semantics to the rdf: and rdfs: Vocabulary

2. D-Entailment: Giving Semantics to Datatypes

3. OWL Entailment: Giving Semantics to the owl: Vocabulary

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

Datatype

A datatype d is an entity characterized by a set of character strings calledlexical forms (or lexical space) and a mapping from that set to a set ofvalues (called value space). Exactly how these sets and mappings (calledlexical-to-value mapping L2V (d)) are defined is a matter external toRDF, e.g. XML Schema [XML Schema Datatypes, 2001].

A datatype map is a set of datatypes.

Besides rdf:XMLLiteral, the RDF spec suggests to support thefollowing XML Schema datatypes:

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

Datatype

A datatype d is an entity characterized by a set of character strings calledlexical forms (or lexical space) and a mapping from that set to a set ofvalues (called value space). Exactly how these sets and mappings (calledlexical-to-value mapping L2V (d)) are defined is a matter external toRDF, e.g. XML Schema [XML Schema Datatypes, 2001].A datatype map is a set of datatypes.

Besides rdf:XMLLiteral, the RDF spec suggests to support thefollowing XML Schema datatypes:

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

Datatype

A datatype d is an entity characterized by a set of character strings calledlexical forms (or lexical space) and a mapping from that set to a set ofvalues (called value space). Exactly how these sets and mappings (calledlexical-to-value mapping L2V (d)) are defined is a matter external toRDF, e.g. XML Schema [XML Schema Datatypes, 2001].A datatype map is a set of datatypes.

Besides rdf:XMLLiteral, the RDF spec suggests to support thefollowing XML Schema datatypes:

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

D-interpretations

D-interpretation

A D-interpretation is an rdfs-interpretation I defined by the following additional semanticconditions:

In principle, just generalization of the Semantic conditions forrdf:XMLLiteral to other datatypes.Inuitive Reading: fix the interpretation of typed literals according toL2V (d)

That’s it!

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

D-interpretations

D-interpretation

A D-interpretation is an rdfs-interpretation I defined by the following additional semanticconditions:

In principle, just generalization of the Semantic conditions forrdf:XMLLiteral to other datatypes.Inuitive Reading: fix the interpretation of typed literals according toL2V (d)

That’s it!

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

D-entailment (|=D) – Examples

{ s p "2.0"∧∧xsd:decimal. } |=D { s p "2"∧∧xsd:integer. }

{ s p "2"∧∧xsd:integer. } |=D { s p "2.0"∧∧xsd:decimal. }

D-inconsistent graphs:

{ ex:a ex:b "25"∧∧xsd:decimal . ex:b rdfs:range xsd:string .}(because the value spaces of sd:string and xsd:decimal are disjoint)

ex:a ex:p "2.5"∧∧xsd:decimal . ex:p rdfs:range xsd:integer .}(overlapping value spaces, but 2.5 outside xsds:integer’s value space)

{ ex:a ex:b "haha"∧∧xsd:decimal. }

(outside lexical space, i.e. “ill-formed” for xsd:decimal)

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

D-entailment (|=D) – Examples

{ s p "2.0"∧∧xsd:decimal. } |=D { s p "2"∧∧xsd:integer. }

{ s p "2"∧∧xsd:integer. } |=D { s p "2.0"∧∧xsd:decimal. }

D-inconsistent graphs:

{ ex:a ex:b "25"∧∧xsd:decimal . ex:b rdfs:range xsd:string .}(because the value spaces of sd:string and xsd:decimal are disjoint)

ex:a ex:p "2.5"∧∧xsd:decimal . ex:p rdfs:range xsd:integer .}(overlapping value spaces, but 2.5 outside xsds:integer’s value space)

{ ex:a ex:b "haha"∧∧xsd:decimal. }

(outside lexical space, i.e. “ill-formed” for xsd:decimal)

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

Rule-based computation of D-entailment?

Basically, needs an oracle to evaluate L2V. . .

e.g. could be done by a built-in that maps all typed literals in a graph tocompute L2V of their value:

s p L2V (l) . ⇐ s p l∧∧d . for l in the lex. space of d.

Again needs “generalized intermediate triples”(UBL ∪ V s × UBL ∪ V s × UBL ∪ V s ) for intermediatecomputation of closure, where V s is the union of the value spaces of allsupported datatypes.

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Unit 3 – RDF(S) & OWL Semantics 2. D-Entailment: Giving Semantics to Datatypes

Rule-based computation of D-entailment?

Basically, needs an oracle to evaluate L2V. . .

e.g. could be done by a built-in that maps all typed literals in a graph tocompute L2V of their value:

s p L2V (l) . ⇐ s p l∧∧d . for l in the lex. space of d.

Again needs “generalized intermediate triples”(UBL ∪ V s × UBL ∪ V s × UBL ∪ V s ) for intermediatecomputation of closure, where V s is the union of the value spaces of allsupported datatypes.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Unit Outline

1. RDF(S) Entailment: Giving semantics to the rdf: and rdfs: Vocabulary

2. D-Entailment: Giving Semantics to Datatypes

3. OWL Entailment: Giving Semantics to the owl: Vocabulary

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Semantics of the owl:Vocabulary

Not part of the RDF Spec, separate document, for the moment, we will restrictourselves to OWL Version 1 [Patel-Schneider et al., 2004].

OWL adds the possiblity to add some DL axioms (TBox, ABox) to RDF(SHOIN (D)).

OWL defines a rewriting how to write such DL axioms in RDF.

OWL adds the possiblity to add some DL statements to an ontology.

Defines two semantics:

• DL-style model-theoretic semantics, for the syntactic subset of OWLwhich corresponds to DL

• RDF compatible semantics

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL DL in two slides: 1/2Expressing property characteristics:

OWL property axioms as RDF triples DL syntax FOL short representationP rdfs:domain C . > v ∀P−.C ∀x, y.P (x, y) ⊃ C(x)

P rdfs:range C . > v ∀P.C ∀x, y.P (x, y) ⊃ C(y)

P owl:inverseOf P0 . P ≡ P−0 ∀x, y.P (x, y) ≡ P0(y, x)

P rdf:type owl:SymmetricProperty. P ≡ P− ∀x, y.P (x, y) ≡ P (y, x)

P rdf:type owl:FunctionalProperty. > v 6 1P ∀x, y, z.P (x, y) ∧ P (x, z) ⊃ y = z

P rdf:type owl:InverseFunctionalProperty. > v 6 1P− ∀x, y, z.P (x, y) ∧ P (z, y) ⊃ x = z

P rdf:type owl:TransitiveProperty. P+ v P ∀x, y, z.P (x, y) ∧ P (y, z) ⊃ P (x, z)

Expressing complex class descriptions:OWL complex class descriptions∗ DL syntax FOL short representationowl:Thing > x = x

owl:Nothing ⊥ ¬x = x

owl:intersectionOf (C1 . . . Cn) C1 u · · · u Cn C1(x) ∧ · · · ∧ Cn(x)

owl:unionOf (C1 . . . Cn) C1 t · · · t Cn C1(x) ∨ · · · ∨ Cn(x)

owl:complementOf (C) ¬C ¬C(x)

owl:oneOf (o1 . . . on) {o1, . . . , on} x = o1 ∨ · · · ∨ x = onowl:restriction (P owl:someValuesFrom (C)) ∃P.C ∃y.P (x, y) ∧ C(y)

owl:restriction (P owl:allValuesFrom (C)) ∀P.C ∀y.P (x, y) ⊃ C(y)

owl:restriction (P owl:value (o)) ∃P.{o} P (x, o)

owl:restriction (P owl:minCardinality (n)) > nP ∃y1...yn.n∧

k=1

P (x, yk)∧∧i<j

yj 6=yj

owl:restriction (P owl:maxCardinality (n)) 6 nP ∀y1...yn+1.

n+1∧k=1

P (x, yk)⊃∨i<j

yi=yj

∗For reasons of legibility, we use a variant of the OWL abstract syntax [Patel-Schneider et al., 2004] in this table.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL DL in two slides: 1/2Expressing property characteristics:

OWL property axioms as RDF triples DL syntax FOL short representationP rdfs:domain C . > v ∀P−.C ∀x, y.P (x, y) ⊃ C(x)

P rdfs:range C . > v ∀P.C ∀x, y.P (x, y) ⊃ C(y)

P owl:inverseOf P0 . P ≡ P−0 ∀x, y.P (x, y) ≡ P0(y, x)

P rdf:type owl:SymmetricProperty. P ≡ P− ∀x, y.P (x, y) ≡ P (y, x)

P rdf:type owl:FunctionalProperty. > v 6 1P ∀x, y, z.P (x, y) ∧ P (x, z) ⊃ y = z

P rdf:type owl:InverseFunctionalProperty. > v 6 1P− ∀x, y, z.P (x, y) ∧ P (z, y) ⊃ x = z

P rdf:type owl:TransitiveProperty. P+ v P ∀x, y, z.P (x, y) ∧ P (y, z) ⊃ P (x, z)

Expressing complex class descriptions:OWL complex class descriptions∗ DL syntax FOL short representationowl:Thing > x = x

owl:Nothing ⊥ ¬x = x

owl:intersectionOf (C1 . . . Cn) C1 u · · · u Cn C1(x) ∧ · · · ∧ Cn(x)

owl:unionOf (C1 . . . Cn) C1 t · · · t Cn C1(x) ∨ · · · ∨ Cn(x)

owl:complementOf (C) ¬C ¬C(x)

owl:oneOf (o1 . . . on) {o1, . . . , on} x = o1 ∨ · · · ∨ x = onowl:restriction (P owl:someValuesFrom (C)) ∃P.C ∃y.P (x, y) ∧ C(y)

owl:restriction (P owl:allValuesFrom (C)) ∀P.C ∀y.P (x, y) ⊃ C(y)

owl:restriction (P owl:value (o)) ∃P.{o} P (x, o)

owl:restriction (P owl:minCardinality (n)) > nP ∃y1...yn.n∧

k=1

P (x, yk)∧∧i<j

yj 6=yj

owl:restriction (P owl:maxCardinality (n)) 6 nP ∀y1...yn+1.

n+1∧k=1

P (x, yk)⊃∨i<j

yi=yj

∗For reasons of legibility, we use a variant of the OWL abstract syntax [Patel-Schneider et al., 2004] in this table.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL DL in two slides: 2/2

Relating Class descriptions:

C1 rdfs:subClassOf C2 C1 v C2

C1 owl:equivalentClass C2 C1 ≡ C2

C1 owl:disjointWith C2 C1 u C2 v ⊥

Relating individuals:

o1 owl:sameAs o1 o1 = o2o1 owl:differentFrom o2 o1 6= o2

Let’s look into some examples ofHow to write OWL in RDF.what it meanspossible problems

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL DL in two slides: 2/2

Relating Class descriptions:

C1 rdfs:subClassOf C2 C1 v C2

C1 owl:equivalentClass C2 C1 ≡ C2

C1 owl:disjointWith C2 C1 u C2 v ⊥

Relating individuals:

o1 owl:sameAs o1 o1 = o2o1 owl:differentFrom o2 o1 6= o2

Let’s look into some examples ofHow to write OWL in RDF.what it meanspossible problems

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

How to write OWL in RDF – A simple ontology about reviewers:

Properties: title, isAuthorOf, publishedIn, etc.

Classes: Senior, Paper, Publication, etc.

Relations:• A Publication is a Paper which has been published (subclass + existential

condition on property)• isAuthorOf is the opposite of Dublin Core’s dc:creator Property2• A Senior researcher is a foaf:Person who isAuthorOf 10+ Publications

(subclass + condition on cardinality)• Each item can be publishedIn at most one venue (functional property)

...

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in DL

∃ex :title.> v ex :Paper (i)

∃ex :title−.> v xsd :string (ii)

ex :isAuthorOf−≡ dc:creator (iii)

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)

>v 6 1 ex :publishedIn− (v)

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u (vi)

∃ex :isAuthorOf .ex :Publication

ex :Club100 ≡ foaf :Person u > 100 ex :isAuthorOf (vii)

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)[a owl:Restriction; owl:onProperty ex:title; owl:someValuesFrom owl:Thing]

rdfs:subclassOf ex:Paper .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)

ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)

ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)

ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u

∃ex :isAuthorOf .ex :Publication(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u∃ex :isAuthorOf .ex :Publication

(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

A. Polleres VU 184.729 37/43

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)owl:Thing rdfs:subclassOf [ a owl:restriction; on Property ex:publishedIn; owl:maxCardinality 1 ].

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u∃ex :isAuthorOf .ex :Publication

(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex:publishedIn a owl:FunctionalProperty .

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u∃ex :isAuthorOf .ex :Publication

(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex:publishedIn a owl:FunctionalProperty .

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u∃ex :isAuthorOf .ex :Publication

(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex:publishedIn a owl:FunctionalProperty .

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u∃ex :isAuthorOf .ex :Publication

(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: in OWL/RDF

∃ex :title.> v ex :Paper (i)ex:title rdfs:domain dc:creator .

∃ex :title−.> v xsd :string (ii)ex:title rdfs:range xs:string .

ex :isAuthorOf−≡ dc:creator (iii)ex:isAuthorOf owl:inverseOf dc:creator .

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

>v 6 1 ex :publishedIn− (v)ex:publishedIn a owl:FunctionalProperty .

ex :Senior≡ foaf :Person u > 10 ex :isAuthorOf u∃ex :isAuthorOf .ex :Publication

(vi)

ex:Senior owl:intersectionOf ( foaf:Person [ a owl:Restriction; owl:onProperty ex:isAuthorOf ;

owl:minCardinality 10 ] [ a owl:Restriction; owl:onProperty ex:isAuthorOf ; owl:someValuesFrom

ex:Publication ] ) .

(vii) Left for exercises!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: What does it mean?

As we have seen, all these axioms boil down to first-order formulas, e.g.

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

Amounts to:

∀x.Publication(x) ≡ Paper(x) ∧ ∃y.publishedIn(x, y)

Recall, so far, when we wrote first-order, we always used a predicate triple:

∀x.triples(x, a, ex:Publication) ≡triples(x, a, ex:Publication)∧∃ytriple(x, ex:publishedIn, y)

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

OWL Example: A simple ontology about reviewers: What does it mean?

As we have seen, all these axioms boil down to first-order formulas, e.g.

ex :Publication≡ ex :Paper u ∃ex :publishedIn.> (iv)ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

Amounts to:

∀x.Publication(x) ≡ Paper(x) ∧ ∃y.publishedIn(x, y)

Recall, so far, when we wrote first-order, we always used a predicate triple:

∀x.triples(x, a, ex:Publication) ≡triples(x, a, ex:Publication)∧∃ytriple(x, ex:publishedIn, y)

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems (differences between OWL DL and OWL Full):

classes/properties used as instances and vice versa (called“meta-modelling”)“faulty” OWL/RDF

⇒ Strictly speaking, if any of this occurs, I can’t use my DL reasoneranymore, or at least it will be incomplete. :-(

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems – faulty OWL/RDF

ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;owl:minCardinality 1 ] ) .

Has no meaning in the DL reading! One could just ignore such statements.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems – faulty OWL/RDF

ex:Publication owl:intersectionOf __ :b1 . __ :b1 rdfs:first __ :b1 .

Has no meaning in the DL reading! One could just ignore suchstatements.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems – “meta-modelling”

ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

exPaper owl:sameAs ex:publishedIn .

Unary/Binary first-order translation doesn’t work anymore:

∀x.Publication(x) ≡ Paper(x) ∧ ∃ypublishedIn(x, y)

∧Publication = publishedIn

Ouch! This is no longer first-order! And outside OWL DL...triple encoding somewhat more stable here:

∀x.triples(x, a, ex:Publication) ≡triples(x, a, ex:Publication) ∧ ∃ytriple(x, ex:publishedIn, y)∧triple(ex:Publication, owl:sameAs, ex:publishedIn)

With the latter, I could still use a first-order reasoner for OWL Full.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems – “meta-modelling”

ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

exPaper owl:sameAs ex:publishedIn .

Unary/Binary first-order translation doesn’t work anymore:

∀x.Publication(x) ≡ Paper(x) ∧ ∃ypublishedIn(x, y)

∧Publication = publishedIn

Ouch! This is no longer first-order! And outside OWL DL...triple encoding somewhat more stable here:

∀x.triples(x, a, ex:Publication) ≡triples(x, a, ex:Publication) ∧ ∃ytriple(x, ex:publishedIn, y)∧triple(ex:Publication, owl:sameAs, ex:publishedIn)

With the latter, I could still use a first-order reasoner for OWL Full.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems – “meta-modelling”

ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

exPaper owl:sameAs ex:publishedIn .

Unary/Binary first-order translation doesn’t work anymore:

∀x.Publication(x) ≡ Paper(x) ∧ ∃ypublishedIn(x, y)

∧Publication = publishedIn

Ouch! This is no longer first-order! And outside OWL DL...

triple encoding somewhat more stable here:

∀x.triples(x, a, ex:Publication) ≡triples(x, a, ex:Publication) ∧ ∃ytriple(x, ex:publishedIn, y)∧triple(ex:Publication, owl:sameAs, ex:publishedIn)

With the latter, I could still use a first-order reasoner for OWL Full.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Possible Problems – “meta-modelling”

ex:Publication owl:intersectionOf ( ex:Paper [ a owl:Restriction; owl:onProperty ex:publishedIn ;

owl:minCardinality 1 ] ) .

exPaper owl:sameAs ex:publishedIn .

Unary/Binary first-order translation doesn’t work anymore:

∀x.Publication(x) ≡ Paper(x) ∧ ∃ypublishedIn(x, y)

∧Publication = publishedIn

Ouch! This is no longer first-order! And outside OWL DL...triple encoding somewhat more stable here:

∀x.triples(x, a, ex:Publication) ≡triples(x, a, ex:Publication) ∧ ∃ytriple(x, ex:publishedIn, y)∧triple(ex:Publication, owl:sameAs, ex:publishedIn)

With the latter, I could still use a first-order reasoner for OWL Full.

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Can I do OWL rules-based FWD-chaining inference as before?

Can I do OWL rules-based FWD-chaining inference as before?with some caveats, yes. . .inherently incomplete. . .. . . but some interesting subset of sound inferences might be covered

[ter Horst, 2005],[Hogan et al., 2009] . . . and last, but not least: OWL 2RL

Can I do OWL use other scalable techniques such as query rewriting?Yes! OWL 2 QL

→ More on that in Lecture 6 (OWL2 and OWL2 profiles)!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Can I do OWL rules-based FWD-chaining inference as before?

Can I do OWL rules-based FWD-chaining inference as before?with some caveats, yes. . .inherently incomplete. . .. . . but some interesting subset of sound inferences might be covered

[ter Horst, 2005],[Hogan et al., 2009] . . . and last, but not least: OWL 2RL

Can I do OWL use other scalable techniques such as query rewriting?Yes! OWL 2 QL

→ More on that in Lecture 6 (OWL2 and OWL2 profiles)!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Can I do OWL rules-based FWD-chaining inference as before?

Can I do OWL rules-based FWD-chaining inference as before?with some caveats, yes. . .inherently incomplete. . .. . . but some interesting subset of sound inferences might be covered

[ter Horst, 2005],[Hogan et al., 2009] . . . and last, but not least: OWL 2RL

Can I do OWL use other scalable techniques such as query rewriting?Yes! OWL 2 QL

→ More on that in Lecture 6 (OWL2 and OWL2 profiles)!

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Unit 3 – RDF(S) & OWL Semantics 3. OWL Entailment: Giving Semantics to the owl: Vocabulary

Recommended Reading

[ter Horst, 2005], RDF, RDFS and parts of OWL entailment in Rules.[Muñoz et al., 2007], minimal sets of complete inference rules forparts of the RDFS vocabulary.

A bit more tough reading (specs), but also recommended:[Hayes, 2004, Sections 3 onwards], official RDF semanticsspecification.

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Unit 3 – RDF(S) & OWL Semantics 4. References

Jos de Bruijn and Stijn Heymans.Logical foundations of (e)RDF(S): Complexity and reasoning.In Proceedings of the 6th International Semantic Web Conference and 2nd AsianSemantic Web Conference (ISWC2007+ASWC2007), number 4825 in Lecture Notes inComputer Science, pages 86–99, Busan, Korea, November 2007. Springer.

P. Hayes.RDF semantics, 2004.http://www.w3.org/TR/rdf-mt/.

Aidan Hogan, Andreas Harth, and Axel Polleres.Scalable authoritative owl reasoning for the web.International Journal on Semantic Web and Information Systems (IJSWIS), 5(2):49–90,2009.

Giovambattista Ianni, Alessandra Martello, Claudio Panetta, and Giorgio Terracina.Efficiently querying RDF(S) ontologies with Answer Set Programming.Journal of Logic and Computation (Special issue), 2009.Forthcoming.

Sergio Muñoz, Jorge Pérez, and Claudio Gutiérrez.Minimal deductive systems for rdf.In ESWC, pages 53–67, 2007.

Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks.OWL Web Ontology Language Semantics and Abstract Syntax, February 2004.W3C Recommendation.

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Unit 3 – RDF(S) & OWL Semantics 4. References

Herman J. ter Horst.Completeness, decidability and complexity of entailment for rdf schema and a semanticextension involving the owl vocabulary.Journal of Web Semantics, 3(2), July 2005.

Xml schema part 2: Datatypes, May 2001.W3C Recommendation, available at http://www.w3.org/TR/xmlschema-2/.

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